Answer to Question #139015 in Classical Mechanics for Max

Question

What is partition function for n dimensional quantum harmonic oscillator.

Answer 1

1-d quantum oscillator is given by a hamiltonian:

$\widehat H_1 = h\omega_1(\widehat a^+a+\frac1 2)$

n-d quantum oscillator :

$\widehat H_N = \sum_{k=1}^{N}h\omega_1(\widehat a_k^+a_k+\frac1 2)$

The partition function of such model is:

$Z_N(\beta) = Tr_{\mathscr{H}}(e^{-\beta\widehat H_N})$

there is $\mathscr{H} -$ hilbert space of n - d quantum oscillator

Rewrite:

$Tr_{\mathscr{H}}(e^{-\beta\widehat H_N}) = \prod^{N}_{k=0}\sum^{\infty}_{n_k=0} e^{-\beta h \omega_k(n_k+\frac 1 2)} =$

$\prod^{N}_{k=0} e^ {\frac {-\beta h \omega_k}{2}} \frac{1}{1-e^ {{-\beta h \omega_k}}} = \frac {2^N}{\prod^{N}_{k=0} sinh(\frac {\beta h \omega_k}{2})}$